我希望我的数学文档能得到专家的检查

Question

在下面的文件中，我尝试举例说明@leandriis 和 David Carlisle 的建议和评论。我只保留了原始文件中的前两个示例。我还做了一些其他修改（简化了一些代码并尝试引入在编辑数学作业或其他内容时可能需要的 LaTeX 对象）。

看看(La)TeX 及其它初学者最常犯的错误是什么？以及https://thinkscience.co.jp/en/articles/LaTeX-habits-to-avoid。

我并不认为自己是专家，但我会给出两点个人意见（我想不是那么原创）：

（LaTeX）尽量将样式（主要是前言部分）与内容分开，让 LaTeX 来做格式化工作。毕竟，它是主要的排版工具。
（数学）在示例或练习的末尾使用小方块（为什么在这里结束列表）是一个品味问题，并引发了长时间的讨论。这个符号的原始意义是指示一个证明刚刚结束。也许它应该保持这样。（LaTeX 会自动在示例环境表示结束。对于此事件，两个信号太多了；就像在短语末尾使用多个感叹号一样。）

\documentclass[12pt, a4paper]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{tikz}
\usepackage{enumitem}


\newtheorem{thm}{Theorem}[section]
\theoremstyle{definition}
\newtheorem{exa}[thm]{Example}
\newtheorem{exe}[thm]{Exercise}
\newtheorem{rem}[thm]{Remark}
\newtheorem*{rem*}{Remark}

\newcommand{\textq}[1]{\text{\quad#1\quad}}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
\newcommand{\sgn}{\operatorname{sgn}}

% \renewcommand{\qed}{\tag*{$\blacksquare$}}
% \newcommand*{\QEDA}{\null\nobreak\hfill\ensuremath{\blacksquare}}
% \newcommand*{\QEDB}{\null\nobreak\hfill\ensuremath{\square}}
% \pagestyle{empty}


\title{Functions and their graphes}
\author{Andr\'e C.}

\begin{document}
\maketitle


\section{Examples and exercises}


\begin{exa}
The \emph{floor} function $y=\floor{x}$ is graphed in
Fig.\,\ref{fig:intPart}.  It is discontinuous at every integer because
the left-hand and right-hand limits are not equal as $x\to n$,
$n\in\mathbb{Z}$:
\[
  \lim_{x\to n^-}\floor{x}=n-1
  \textq{and}
  \lim_{x\to n^+}\floor{x}=n.
\]
We see that, since $\lfloor n\rfloor=n$, the floor function is
right-continuous at every integer $n$ (but not left-continuous). 

The floor function is continuous at every real number other
than the integers.  For example,
$\lim_{x\to1.5}\floor{x}=1=\floor{1.5}$.
In general, if $n-1<c<n$, $n$ an integer, then
\[
  \lim_{x\to c}\floor{x}=n-1=\floor{c}.
\]
\begin{figure}[ht!]
  \centering
  \begin{tikzpicture}[scale=.8, every node/.style={scale=.8}]
    \draw[->] (-2,0)--(5,0) node[right]{$x$};
    \draw[->] (0,-3)--(0,5) node[above]{$y$};
    
    \foreach \x in {-1, 1, 2, 3, 4} \draw (\x,0) -- +(0,-3pt)
    node[below=3pt]{$\x$};
    \foreach \y in {-2,1,2,3,4} \draw (0,\y) -- +(-3pt,0)
    node[left=3pt]{$\y$};

    \begin{scope}[blue!50!green, thick]
      \draw (-2,-2) -- (-1,-2);
      \draw[fill=white] (-1,-2) circle (2pt);
      
      \foreach \x in {-1, 1, 2, 3}{%
        \filldraw (\x, \x) circle (2pt)
        -- +(1, 0) circle (2pt) node[circle, fill=white, inner sep=1.4pt] {};
      }
    
      \filldraw (4,4) circle (2pt) -- (5,4);
    \end{scope}

  \end{tikzpicture}
  \caption{The graph $y=\floor{x}$.  The floor function is continuous
    at every non-integer point and it is right-continuous, but not
    left-continuous otherwise.}
  \label{fig:intPart}
\end{figure}
\end{exa}




\begin{rem*}
Note that the \emph{ceiling} function is defined by
$x\mapsto\ceil{x}=\min\{n\in\mathbb{Z} \mid n\geq x\}$.  As for the
\emph{integer part} function, its definition varies sometimes; I don't
know if it's culturally dependent\ldots\ It can be defined either by
$E(x)=\floor{x}$, especially in the Latin culture, or by
$E(x)=\sgn{x}\cdot\floor{|x|}$, in connection with the computer science
languages.
\end{rem*}


\begin{exe}
  \label{e:graph}
Find the horizontal and vertical asymptotes of the graph of
\[
  f(x)=-\frac{8}{x^2-4}.
\]

\noindent
\emph{Solution}.
We are interested in the behavior of the graph as $x\to\pm\infty$ and
as $x\to\pm2$---where the denominator is zero.  Notice that $f$ is an
even function of $x$, so its graph is symmetric with respect to the
$y$-axis.
\begin{enumerate}[wide, labelindent=0pt, label={\textbf{(\alph*)}}]
\item
\emph{The behavior as $x\to\pm\infty$}. Since
$\lim_{x\to\infty}f(x)=0$, the line $y=0$ is a horizontal asymptote of
the graph to the right. By symmetry it is an asymptote to the left as
well (Figure 0.2). Notice that the curve approaches the $x$-axis from
only the negative side (or from below). Also, $f(0)=2$.
\item
\emph{The behavior as $x\to\pm2$}. Since
\[
  \lim_{x\to2^+}f(x)=-\infty
  \textq{and}
  \lim_{x\to2^-}f(x)=\infty,
\]
the line $x=2$ is a vertical asymptote bith from the right and from
the left. By symmetry, the line $x=-2$ is also a vertical asymptote.
\end{enumerate}

There are no other asymptotes because $f$ has a finite limit at every
other point.  The graph is shown in Fig.\,\ref{fig:graph}.
\qed
\end{exe}


\begin{figure}[ht!]
  \centering
  \begin{tikzpicture}[every node/.style={scale=.8}]
    \draw[->, thin] (-7,0)--(7,0) node[right]{$x$};
    \draw[->, thin] (0,-4.5)--(0,9) node[above]{$y$};
    
    \draw[color=red](2,8)--(2,-4.3);
    \draw[color=red](-2,8)--(-2,-4.3);
    \draw[color=red](-5.5,0)--(5.5,0);
    
    \foreach \x in {-4,-3,-2,-1, 0, 1, 2, 3, 4}
    \draw (\x,0)--(\x,2pt) node[below=3pt, fill=white]{$\x$};
    \foreach \y in {1,2,3,4,5,6,7,8}
    \draw (0,\y)--(2pt,\y) node[left=3pt]{$\y$};
    \foreach \y in {-4,-3,-2,-1} \draw (0,\y)--(2pt,\y);
   
    \draw[color=blue!50!green, thick]
    (-1.9,8) .. controls (-1.88,6) and (-1.7,2.1) .. (0,2);
    \draw[color=blue!50!green, thick]
    (1.9,8) .. controls (1.88,6) and (1.7,2.1) .. (0,2);
    
    \draw[color=blue!50!green, thick]
    (5.5, -0.2) .. controls (3, -0.3) and (2.3, -1) .. (2.2, -4.3);
    \draw[color=blue!50!green, thick]
    (-5.5, -0.2) .. controls (-3, -0.3) and (-2.3, -1) .. (-2.2, -4.3);
    
    \draw (2,4) node[right]{vertical asymptote, $x=2$};
    \draw (-2,4) node[left]{vertical asymptote, $x=-2$};
    \draw[gray, thin] (2.5, 2pt) to[out=80, in=180] (3, 0.8)
    node[text=black, right] {horizontal asymptote, $y=0$};
\end{tikzpicture}
  \caption{Graph of the function $y=-\dfrac{8}{x^2-4}$.  Notice
    that the curve approaches the $x$-axis from only one side.
    Asymptotes do not have to be two sided.}
  \label{fig:graph}
\end{figure}

\end{document}

Answer 1

在下面的文件中，我尝试举例说明@leandriis 和 David Carlisle 的建议和评论。我只保留了原始文件中的前两个示例。我还做了一些其他修改（简化了一些代码并尝试引入在编辑数学作业或其他内容时可能需要的 LaTeX 对象）。

看看(La)TeX 及其它初学者最常犯的错误是什么？以及https://thinkscience.co.jp/en/articles/LaTeX-habits-to-avoid。

我并不认为自己是专家，但我会给出两点个人意见（我想不是那么原创）：

（LaTeX）尽量将样式（主要是前言部分）与内容分开，让 LaTeX 来做格式化工作。毕竟，它是主要的排版工具。
（数学）在示例或练习的末尾使用小方块（为什么在这里结束列表）是一个品味问题，并引发了长时间的讨论。这个符号的原始意义是指示一个证明刚刚结束。也许它应该保持这样。（LaTeX 会自动在示例环境表示结束。对于此事件，两个信号太多了；就像在短语末尾使用多个感叹号一样。）

\documentclass[12pt, a4paper]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{tikz}
\usepackage{enumitem}


\newtheorem{thm}{Theorem}[section]
\theoremstyle{definition}
\newtheorem{exa}[thm]{Example}
\newtheorem{exe}[thm]{Exercise}
\newtheorem{rem}[thm]{Remark}
\newtheorem*{rem*}{Remark}

\newcommand{\textq}[1]{\text{\quad#1\quad}}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
\newcommand{\sgn}{\operatorname{sgn}}

% \renewcommand{\qed}{\tag*{$\blacksquare$}}
% \newcommand*{\QEDA}{\null\nobreak\hfill\ensuremath{\blacksquare}}
% \newcommand*{\QEDB}{\null\nobreak\hfill\ensuremath{\square}}
% \pagestyle{empty}


\title{Functions and their graphes}
\author{Andr\'e C.}

\begin{document}
\maketitle


\section{Examples and exercises}


\begin{exa}
The \emph{floor} function $y=\floor{x}$ is graphed in
Fig.\,\ref{fig:intPart}.  It is discontinuous at every integer because
the left-hand and right-hand limits are not equal as $x\to n$,
$n\in\mathbb{Z}$:
\[
  \lim_{x\to n^-}\floor{x}=n-1
  \textq{and}
  \lim_{x\to n^+}\floor{x}=n.
\]
We see that, since $\lfloor n\rfloor=n$, the floor function is
right-continuous at every integer $n$ (but not left-continuous). 

The floor function is continuous at every real number other
than the integers.  For example,
$\lim_{x\to1.5}\floor{x}=1=\floor{1.5}$.
In general, if $n-1<c<n$, $n$ an integer, then
\[
  \lim_{x\to c}\floor{x}=n-1=\floor{c}.
\]
\begin{figure}[ht!]
  \centering
  \begin{tikzpicture}[scale=.8, every node/.style={scale=.8}]
    \draw[->] (-2,0)--(5,0) node[right]{$x$};
    \draw[->] (0,-3)--(0,5) node[above]{$y$};
    
    \foreach \x in {-1, 1, 2, 3, 4} \draw (\x,0) -- +(0,-3pt)
    node[below=3pt]{$\x$};
    \foreach \y in {-2,1,2,3,4} \draw (0,\y) -- +(-3pt,0)
    node[left=3pt]{$\y$};

    \begin{scope}[blue!50!green, thick]
      \draw (-2,-2) -- (-1,-2);
      \draw[fill=white] (-1,-2) circle (2pt);
      
      \foreach \x in {-1, 1, 2, 3}{%
        \filldraw (\x, \x) circle (2pt)
        -- +(1, 0) circle (2pt) node[circle, fill=white, inner sep=1.4pt] {};
      }
    
      \filldraw (4,4) circle (2pt) -- (5,4);
    \end{scope}

  \end{tikzpicture}
  \caption{The graph $y=\floor{x}$.  The floor function is continuous
    at every non-integer point and it is right-continuous, but not
    left-continuous otherwise.}
  \label{fig:intPart}
\end{figure}
\end{exa}




\begin{rem*}
Note that the \emph{ceiling} function is defined by
$x\mapsto\ceil{x}=\min\{n\in\mathbb{Z} \mid n\geq x\}$.  As for the
\emph{integer part} function, its definition varies sometimes; I don't
know if it's culturally dependent\ldots\ It can be defined either by
$E(x)=\floor{x}$, especially in the Latin culture, or by
$E(x)=\sgn{x}\cdot\floor{|x|}$, in connection with the computer science
languages.
\end{rem*}


\begin{exe}
  \label{e:graph}
Find the horizontal and vertical asymptotes of the graph of
\[
  f(x)=-\frac{8}{x^2-4}.
\]

\noindent
\emph{Solution}.
We are interested in the behavior of the graph as $x\to\pm\infty$ and
as $x\to\pm2$---where the denominator is zero.  Notice that $f$ is an
even function of $x$, so its graph is symmetric with respect to the
$y$-axis.
\begin{enumerate}[wide, labelindent=0pt, label={\textbf{(\alph*)}}]
\item
\emph{The behavior as $x\to\pm\infty$}. Since
$\lim_{x\to\infty}f(x)=0$, the line $y=0$ is a horizontal asymptote of
the graph to the right. By symmetry it is an asymptote to the left as
well (Figure 0.2). Notice that the curve approaches the $x$-axis from
only the negative side (or from below). Also, $f(0)=2$.
\item
\emph{The behavior as $x\to\pm2$}. Since
\[
  \lim_{x\to2^+}f(x)=-\infty
  \textq{and}
  \lim_{x\to2^-}f(x)=\infty,
\]
the line $x=2$ is a vertical asymptote bith from the right and from
the left. By symmetry, the line $x=-2$ is also a vertical asymptote.
\end{enumerate}

There are no other asymptotes because $f$ has a finite limit at every
other point.  The graph is shown in Fig.\,\ref{fig:graph}.
\qed
\end{exe}


\begin{figure}[ht!]
  \centering
  \begin{tikzpicture}[every node/.style={scale=.8}]
    \draw[->, thin] (-7,0)--(7,0) node[right]{$x$};
    \draw[->, thin] (0,-4.5)--(0,9) node[above]{$y$};
    
    \draw[color=red](2,8)--(2,-4.3);
    \draw[color=red](-2,8)--(-2,-4.3);
    \draw[color=red](-5.5,0)--(5.5,0);
    
    \foreach \x in {-4,-3,-2,-1, 0, 1, 2, 3, 4}
    \draw (\x,0)--(\x,2pt) node[below=3pt, fill=white]{$\x$};
    \foreach \y in {1,2,3,4,5,6,7,8}
    \draw (0,\y)--(2pt,\y) node[left=3pt]{$\y$};
    \foreach \y in {-4,-3,-2,-1} \draw (0,\y)--(2pt,\y);
   
    \draw[color=blue!50!green, thick]
    (-1.9,8) .. controls (-1.88,6) and (-1.7,2.1) .. (0,2);
    \draw[color=blue!50!green, thick]
    (1.9,8) .. controls (1.88,6) and (1.7,2.1) .. (0,2);
    
    \draw[color=blue!50!green, thick]
    (5.5, -0.2) .. controls (3, -0.3) and (2.3, -1) .. (2.2, -4.3);
    \draw[color=blue!50!green, thick]
    (-5.5, -0.2) .. controls (-3, -0.3) and (-2.3, -1) .. (-2.2, -4.3);
    
    \draw (2,4) node[right]{vertical asymptote, $x=2$};
    \draw (-2,4) node[left]{vertical asymptote, $x=-2$};
    \draw[gray, thin] (2.5, 2pt) to[out=80, in=180] (3, 0.8)
    node[text=black, right] {horizontal asymptote, $y=0$};
\end{tikzpicture}
  \caption{Graph of the function $y=-\dfrac{8}{x^2-4}$.  Notice
    that the curve approaches the $x$-axis from only one side.
    Asymptotes do not have to be two sided.}
  \label{fig:graph}
\end{figure}

\end{document}

我希望我的数学文档能得到专家的检查

答案1

相关内容